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			<h1 id="firstHeading" class="firstHeading" lang="en"><span dir="auto">Move-to-front transform</span></h1>
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<td class="mbox-text" style=""><span class="mbox-text-span">This article includes a <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, related reading or <a href="http://en.wikipedia.org/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, but <b>its sources remain unclear because it lacks <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>. <span class="hide-when-compact">Please <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by introducing more precise citations.</span> <small><i>(May 2011)</i></small> </span></td>
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<p>The <b>move-to-front transform</b> (or <b>MTF</b>) is an <a href="http://en.wikipedia.org/wiki/Code" title="Code">encoding</a> of <a href="http://en.wikipedia.org/wiki/Data" title="Data">data</a> (typically a stream of <a href="http://en.wikipedia.org/wiki/Byte" title="Byte">bytes</a>) designed to improve the performance of <a href="http://en.wikipedia.org/wiki/Entropy_encoding" title="Entropy encoding">entropy encoding</a> techniques of <a href="http://en.wikipedia.org/wiki/Data_compression" title="Data compression">compression</a>.
 When efficiently implemented, it is fast enough that its benefits 
usually justify including it as an extra step in data compression <a href="http://en.wikipedia.org/wiki/Algorithm" title="Algorithm">algorithms</a>.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#The_transform"><span class="tocnumber">1</span> <span class="toctext">The transform</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Implementation"><span class="tocnumber">2</span> <span class="toctext">Implementation</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Use_in_practical_data_compression_algorithms"><span class="tocnumber">3</span> <span class="toctext">Use in practical data compression algorithms</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#External_links"><span class="tocnumber">4</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Move-to-front_transform&amp;action=edit&amp;section=1" title="Edit section: The transform">edit</a>]</span> <span class="mw-headline" id="The_transform">The transform</span></h2>
<p>The main idea is that each symbol in the data is replaced by its 
index in the stack of “recently used symbols”. For example, long 
sequences of identical symbols are replaced by as many zeroes, whereas 
when a symbol that has not been used in a long time appears, it is 
replaced with a large number. Thus at the end the data is transformed 
into a sequence of integers; if the data exhibits a lot of local 
correlations, then these integers tend to be small.</p>
<p>This algorithm was published in the following paper: Ryabko, B. Ya. 
Data compression by means of a "book stack”, Problems of Information 
Transmission, 1980, v. 16: (4), pp.&nbsp;265–269. The original name of 
this code is “book stack”. The history of discovery of the book stack 
(or move-to-front) code can be found here: Ryabko, B. Ya.; Horspool, R. 
Nigel; Cormack, Gordon V. Comments to: "A locally adaptive data 
compression scheme" by J. L. Bentley, D. D. Sleator, R. E. Tarjan and V.
 K. Wei. Comm. ACM 30 (1987), no. 9, 792—794.</p>
<p>Let us give a precise description. Assume for simplicity that the symbols in the data are <a href="http://en.wikipedia.org/wiki/Byte" title="Byte">bytes</a>. Each byte value is encoded by its index in a <a href="http://en.wikipedia.org/wiki/List_%28computing%29" title="List (computing)" class="mw-redirect">list</a>
 of bytes, which changes over the course of the algorithm. The list is 
initially in order by byte value (0, 1, 2, 3, ..., 255). Therefore, the 
first byte is always encoded by its own value. However, after encoding a
 byte, that value is moved to the front of the list before continuing to
 the next byte.</p>
<p>An example will shed some light on how the transform works. Imagine 
instead of bytes, we are encoding values in a-z. We wish to transform 
the following sequence:</p>
<pre>bananaaa
</pre>
<p>By convention, the list is initially (abcdefghijklmnopqrstuvwxyz). 
The first letter in the sequence is b, which appears at index 1 (the 
list is indexed from 0 to 25). We put a 1 to the output stream:</p>
<pre>1
</pre>
<p>The b moves to the front of the list, producing 
(bacdefghijklmnopqrstuvwxyz). The next letter is a, which now appears at
 index 1. So we add a 1 to the output stream. We have:</p>
<pre>1,1
</pre>
<p>and we move back the letter a to the top of the list. Continuing this way, we find that the sequence is encoded by:</p>
<pre>1,1,13,1,1,1,0,0
</pre>
<table class="wikitable" border="1">
<tbody><tr>
<th>Iteration</th>
<th>Sequence</th>
<th>List</th>
</tr>
<tr>
<td><b>b</b>ananaaa</td>
<td>1</td>
<td>(abcdefghijklmnopqrstuvwxyz)</td>
</tr>
<tr>
<td>b<b>a</b>nanaaa</td>
<td>1,1</td>
<td>(bacdefghijklmnopqrstuvwxyz)</td>
</tr>
<tr>
<td>ba<b>n</b>anaaa</td>
<td>1,1,13</td>
<td>(abcdefghijklmnopqrstuvwxyz)</td>
</tr>
<tr>
<td>ban<b>a</b>naaa</td>
<td>1,1,13,1</td>
<td>(nabcdefghijklmopqrstuvwxyz)</td>
</tr>
<tr>
<td>bana<b>n</b>aaa</td>
<td>1,1,13,1,1</td>
<td>(anbcdefghijklmopqrstuvwxyz)</td>
</tr>
<tr>
<td>banan<b>a</b>aa</td>
<td>1,1,13,1,1,1</td>
<td>(nabcdefghijklmopqrstuvwxyz)</td>
</tr>
<tr>
<td>banana<b>a</b>a</td>
<td>1,1,13,1,1,1,0</td>
<td>(anbcdefghijklmopqrstuvwxyz)</td>
</tr>
<tr>
<td>bananaa<b>a</b></td>
<td>1,1,13,1,1,1,0,0</td>
<td>(anbcdefghijklmopqrstuvwxyz)</td>
</tr>
<tr>
<td>Final</td>
<td>1,1,13,1,1,1,0,0</td>
<td>(anbcdefghijklmopqrstuvwxyz)</td>
</tr>
</tbody></table>
<p>It is easy to see that the transform is reversible. Simply maintain 
the same list and decode by replacing each index in the encoded stream 
with the letter at that index in the list. Note the difference between 
this and the encoding method: The index in the list is used directly 
instead of looking up each value for its index.</p>
<p>i.e. you start again with (abcdefghijklmnopqrstuvwxyz). You take the 
"1" of the encoded block and look it up in the list, which results in 
"b". Then move the "b" to front which results in (bacdef...). Then take 
the next "1", look it up in the list, this results in "a", move the "a" 
to front ... etc.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Move-to-front_transform&amp;action=edit&amp;section=2" title="Edit section: Implementation">edit</a>]</span> <span class="mw-headline" id="Implementation">Implementation</span></h2>
<p>Details of implementation are important for performance, particularly
 for decoding. For encoding, no clear advantage is gained by using a <a href="http://en.wikipedia.org/wiki/Linked_list" title="Linked list">linked list</a>, so using an <a href="http://en.wikipedia.org/wiki/Array_data_structure" title="Array data structure">array</a> to store the list is acceptable, with worst case performance <a href="http://en.wikipedia.org/wiki/Big_O_notation" title="Big O notation">O</a>(<var>n</var><var>k</var>), where <var>n</var> is the length of the data to be encoded and <var>k</var> is the number of values (generally a constant for a given implementation).</p>
<p>However, for decoding, we can use specialized data structures to greatly improve performance.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs examples for clarification from February 2012">examples needed</span></a></i>]</sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Move-to-front_transform&amp;action=edit&amp;section=3" title="Edit section: Use in practical data compression algorithms">edit</a>]</span> <span class="mw-headline" id="Use_in_practical_data_compression_algorithms">Use in practical data compression algorithms</span></h2>
<p>The MTF transform takes advantage of local correlation of frequencies to reduce the <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29" title="Entropy (information theory)">entropy</a>
 of a message. Indeed, recently used letters stay towards the front of 
the list; if use of letters exhibits local correlations, this will 
result in a large number of small numbers such as "0"'s and "1"'s in the
 output.</p>
<p>However, not all data exhibits this type of local correlation, and 
for some messages, the MTF transform may actually increase the entropy.</p>
<p>An important use of the MTF transform is in <a href="http://en.wikipedia.org/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">Burrows–Wheeler transform</a>
 based compression. The Burrows-Wheeler transform is very good at 
producing a sequence that exhibits local frequency correlation from <a href="http://en.wikipedia.org/wiki/Plain_text" title="Plain text">text</a>
 and certain other special classes of data. Compression benefits greatly
 from following up the Burrows-Wheeler transform with an MTF transform 
before the final entropy-encoding step.</p>
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</td>
<td class="mbox-text" style=""><span class="mbox-text-span">This following section <b>may be <a href="http://en.wikipedia.org/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers</b>. In particular, What is the method used to arrive at these numbers?. <span class="hide-when-compact">Please help us <a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the following section</a>; suggestions may be found on the <a href="http://en.wikipedia.org/wiki/Talk:Move-to-front_transform" title="Talk:Move-to-front transform">talk page</a>.</span> <small><i>(February 2011)</i></small> </span></td>
</tr>
</tbody></table>
<p>As an example, imagine we wish to compress <a href="http://en.wikipedia.org/wiki/To_be,_or_not_to_be" title="To be, or not to be">Hamlet's soliloquy</a> (<i>To be, or not to be...</i>).
 We can calculate the entropy of this message to be 7033 bits. Naively, 
we might try to apply the MTF transform directly. The result is a 
message with 7807 bits of entropy (higher than the original). The reason
 is that English text does not in general exhibit a high level of local 
frequency correlation. However, if we first apply the Burrows-Wheeler 
transform, and then the MTF transform, we get a message with 6187 bits 
of entropy. Note that the Burrows-Wheeler transform does not decrease 
the entropy of the message; it only reorders the bytes in a way that 
makes the MTF transform more effective.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Move-to-front_transform&amp;action=edit&amp;section=4" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a rel="nofollow" class="external text" href="http://www.cs.cmu.edu/%7Esleator/papers/Adaptive-data-compression.htm">"A
 Locally Adaptive Data Compression Scheme" by J. L. Bentley, D. D. 
Sleator, R. E. Tarjan, V. K. Wei, Communications of the ACM-Vol. 29, No.
 4, 1986</a></li>
<li><a rel="nofollow" class="external text" href="http://www.arturocampos.com/ac_mtf.html">"Move to front" by Arturo San Emeterio Campos</a></li>
</ul>


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